3.23.46 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [2246]

Optimal. Leaf size=288 \[ -\frac {(3 c e f-7 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {\sqrt {2 c d-b e} (3 c e f-7 c d g+2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]

[Out]

-1/3*(2*b*e*g-7*c*d*g+3*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e*x+d)^(3/2)-(-d*g+e*f
)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)/(e*x+d)^(7/2)+(2*b*e*g-7*c*d*g+3*c*e*f)*arctanh((d*(
-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2))*(-b*e+2*c*d)^(1/2)/e^2-(2*b*e*g-7*c*d*g+3
*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {806, 678, 674, 214} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-7 c d g+3 c e f)}{3 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-7 c d g+3 c e f)}{e^2 \sqrt {d+e x}}+\frac {\sqrt {2 c d-b e} (2 b e g-7 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

-(((3*c*e*f - 7*c*d*g + 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqrt[d + e*x])) - ((3*c*e*f -
7*c*d*g + 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)) - ((e*f
- d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + (Sqrt[2*c*d - b*e]*
(3*c*e*f - 7*c*d*g + 2*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*
x])])/e^2

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 678

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(3 c e f-7 c d g+2 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {((-2 c d+b e) (3 c e f-7 c d g+2 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(3 c e f-7 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {((2 c d-b e) (3 c e f-7 c d g+2 b e g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e}\\ &=-\frac {(3 c e f-7 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-((2 c d-b e) (3 c e f-7 c d g+2 b e g)) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\frac {(3 c e f-7 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {\sqrt {2 c d-b e} (3 c e f-7 c d g+2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 192, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {(d+e x) (-b e+c (d-e x))} \left (\sqrt {-b e+c (d-e x)} \left (b e (-3 e f+11 d g+8 e g x)+2 c \left (-13 d^2 g+d e (6 f-9 g x)+e^2 x (3 f+g x)\right )\right )+3 \sqrt {-2 c d+b e} (-3 c e f+7 c d g-2 b e g) (d+e x) \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )\right )}{3 e^2 (d+e x)^{3/2} \sqrt {-b e+c (d-e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

-1/3*(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(Sqrt[-(b*e) + c*(d - e*x)]*(b*e*(-3*e*f + 11*d*g + 8*e*g*x) + 2*
c*(-13*d^2*g + d*e*(6*f - 9*g*x) + e^2*x*(3*f + g*x))) + 3*Sqrt[-2*c*d + b*e]*(-3*c*e*f + 7*c*d*g - 2*b*e*g)*(
d + e*x)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]]))/(e^2*(d + e*x)^(3/2)*Sqrt[-(b*e) + c*(d - e*x
)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(266)=532\).
time = 0.04, size = 687, normalized size = 2.39

method	result	size
default	\(\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x -33 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x +9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x +42 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -2 c \,e^{2} g \,x^{2} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g -33 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g +9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f +42 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f -8 b \,e^{2} g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+18 c d e g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-6 c \,e^{2} f x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-11 b d e g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+3 b \,e^{2} f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+26 c \,d^{2} g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-12 c d e f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {-c e x -b e +c d}\, e^{2} \sqrt {b e -2 c d}}\)	\(687\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*e^3*g*x-33*arctan
((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*g*x+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*
c*e^3*f*x+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(
b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x-2*c*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+6*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*d*e^2*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g+9*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*f+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c
^2*d^3*g-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f-8*b*e^2*g*x*(b*e-2*c*d)^(1/2)*(-c*e*x
-b*e+c*d)^(1/2)+18*c*d*e*g*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-6*c*e^2*f*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*
e+c*d)^(1/2)-11*b*d*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+3*b*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^
(1/2)+26*c*d^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-12*c*d*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2))
/(e*x+d)^(3/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*(g*x + f)/(x*e + d)^(7/2), x)

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Fricas [A]
time = 2.71, size = 566, normalized size = 1.97 \begin {gather*} \left [-\frac {3 \, {\left (7 \, c d^{3} g - {\left (3 \, c f + 2 \, b g\right )} x^{2} e^{3} + {\left (7 \, c d g x^{2} - 2 \, {\left (3 \, c d f + 2 \, b d g\right )} x\right )} e^{2} + {\left (14 \, c d^{2} g x - 3 \, c d^{2} f - 2 \, b d^{2} g\right )} e\right )} \sqrt {2 \, c d - b e} \log \left (\frac {3 \, c d^{2} - {\left (c x^{2} + 2 \, b x\right )} e^{2} + 2 \, {\left (c d x - b d\right )} e + 2 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {2 \, c d - b e} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (26 \, c d^{2} g - {\left (2 \, c g x^{2} - 3 \, b f + 2 \, {\left (3 \, c f + 4 \, b g\right )} x\right )} e^{2} + {\left (18 \, c d g x - 12 \, c d f - 11 \, b d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{6 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{2}\right )}}, -\frac {3 \, {\left (7 \, c d^{3} g - {\left (3 \, c f + 2 \, b g\right )} x^{2} e^{3} + {\left (7 \, c d g x^{2} - 2 \, {\left (3 \, c d f + 2 \, b d g\right )} x\right )} e^{2} + {\left (14 \, c d^{2} g x - 3 \, c d^{2} f - 2 \, b d^{2} g\right )} e\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-2 \, c d + b e} \sqrt {x e + d}}{\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}\right ) - {\left (26 \, c d^{2} g - {\left (2 \, c g x^{2} - 3 \, b f + 2 \, {\left (3 \, c f + 4 \, b g\right )} x\right )} e^{2} + {\left (18 \, c d g x - 12 \, c d f - 11 \, b d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(7*c*d^3*g - (3*c*f + 2*b*g)*x^2*e^3 + (7*c*d*g*x^2 - 2*(3*c*d*f + 2*b*d*g)*x)*e^2 + (14*c*d^2*g*x -
3*c*d^2*f - 2*b*d^2*g)*e)*sqrt(2*c*d - b*e)*log((3*c*d^2 - (c*x^2 + 2*b*x)*e^2 + 2*(c*d*x - b*d)*e + 2*sqrt(c*
d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(2*c*d - b*e)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(26*c*d^2*g -
(2*c*g*x^2 - 3*b*f + 2*(3*c*f + 4*b*g)*x)*e^2 + (18*c*d*g*x - 12*c*d*f - 11*b*d*g)*e)*sqrt(c*d^2 - b*d*e - (c
*x^2 + b*x)*e^2)*sqrt(x*e + d))/(x^2*e^4 + 2*d*x*e^3 + d^2*e^2), -1/3*(3*(7*c*d^3*g - (3*c*f + 2*b*g)*x^2*e^3
+ (7*c*d*g*x^2 - 2*(3*c*d*f + 2*b*d*g)*x)*e^2 + (14*c*d^2*g*x - 3*c*d^2*f - 2*b*d^2*g)*e)*sqrt(-2*c*d + b*e)*a
rctan(-sqrt(-2*c*d + b*e)*sqrt(x*e + d)/sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)) - (26*c*d^2*g - (2*c*g*x^2 -
3*b*f + 2*(3*c*f + 4*b*g)*x)*e^2 + (18*c*d*g*x - 12*c*d*f - 11*b*d*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^
2)*sqrt(x*e + d))/(x^2*e^4 + 2*d*x*e^3 + d^2*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**(7/2), x)

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Giac [A]
time = 0.89, size = 346, normalized size = 1.20 \begin {gather*} \frac {{\left (18 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} d g - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} f e - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c g e + 2 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c g + \frac {3 \, {\left (14 \, c^{3} d^{2} g - 6 \, c^{3} d f e - 11 \, b c^{2} d g e + 3 \, b c^{2} f e^{2} + 2 \, b^{2} c g e^{2}\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e + \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2}\right )}}{{\left (x e + d\right )} c}\right )} e^{\left (-2\right )}}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

1/3*(18*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^2*d*g - 6*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^2*f*e - 6*sqrt(-(x*e +
d)*c + 2*c*d - b*e)*b*c*g*e + 2*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*g + 3*(14*c^3*d^2*g - 6*c^3*d*f*e - 11*b
*c^2*d*g*e + 3*b*c^2*f*e^2 + 2*b^2*c*g*e^2)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-
2*c*d + b*e) + 3*(2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^3*d^2*g - 2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^3*d*f*e
- sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c^2*d*g*e + sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c^2*f*e^2)/((x*e + d)*c))*
e^(-2)/c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^(7/2),x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^(7/2), x)

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